Construction et classification de certaines solutions algébriques des systèmes de Garnier

22 pagesInternational audienceIn this paper, we classify all (complete) non elementary algebraic solutions of Garnier systems that can be constructed by Kitaev's method: they are deduced from isomonodromic deformations defined by pulling back a given fuchsian equation E by a family of ramified covers. We first introduce orbifold structures associated to a fuchsian equation. This allow to get a refined version of Riemann-Hurwitz formula and then to promtly deduce that E is hypergeometric. Then, we can bound exponents and degree of the pull-back maps and further list all possible ramification cases. This generalizes a result due to C. Doran for the Painleve VI case. We explicitely construct one of these solutions

Galois Covers, Grothendieck-Teichmüller Theory and Dessins d’Enfants - An Introduction

In this introduction, we will give a brief overview of the themes and topics of the articles in this proceedings volume and summarise each individual contribution based on the abstracts and introduction. The LMS workshop Galois covers, Grothen dieck-Teichmuller theory and Dessins d’enfants brought together many experts from the United Kingdom and abroad as well as graduate students and early career researchers centred around three main themes: Galois covers, Grothendieck-Teichmuller theory and dessins d’enfants. These themes bring about unexpected links between algebraic geometry, representation theory, number theory and algebraic topology. The proceedings of this workshop reflect the topics and scope of the lectures presented and explore through surveys and original research articles the many facets of these fascinating themes